Slowly rotating gas-rich galaxies in modified Newtonian dynamics (MOND)

0.21 MB·English

by  F. J. Sanchez-Salcedo

Draftversion January25,2013
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
SLOWLY ROTATING GAS-RICH GALAXIES IN MODIFIED NEWTONIAN DYNAMICS (MOND)
F. J. Sa´nchez-Salcedo1, A. M. Hidalgo-Ga´mez2 and Eric E. Mart´ınez-Garc´ıa1
Draft version January 25, 2013
ABSTRACT
We have carriedout a searchfor gas-richdwarf galaxies that have lower rotation velocities in their
outskirtsthan MOdifiedNewtonian Dynamics (MOND) predicts, sothat the amplitude oftheir rota-
3
tion curves cannot be fitted by arbitrarily increasing the mass-to-light ratio of the stellar component
1
0 or by assuming additional undetected matter. With presently available data, the gas-rich galaxies
2 UGC 4173, Holmberg II, ESO 245-G05, NGC 4861 and ESO 364-G029 deviate most from MOND
predictions and, thereby, provide a sample of promising targets in testing the MOND framework. In
n
the case of Holmberg II and NGC 4861, we find that their rotation curves are probably inconsistent
a
with MOND, unless their inclinations and distances differ significantly from the nominal ones. The
J
galaxy ESO 364-G029 is a promising target because its baryonic mass and rotation curve are similar
4
to Holmberg II but presents a higher inclination. Deeper photometric and Hi observations of ESO
2
364-G029, together with further decreasing systematic uncertainties, may provide a strong test to

MOND.
]
O Subject headings: galaxies: dwarf – galaxies: kinematics and dynamics – dark matter – gravitation
C
. 1. INTRODUCTION freeparameters[butseealsoForeman&Scott(2011)and
h
the reply by McGaugh (2011)]. However, we must note
p Milgrom (1983) proposed that a modification of New-
thatitwasalreadyknownatthattime that the rotation
- ton gravitational law at accelerations below a threshold
o of 10−8 cm s−2, could explain the dynamics of galax- curvesof22ofthegalaxiesinMcGaugh’ssample(47%of
r ≈ the galaxies)wereinagreementwithMONDpredictions
t ies without invoking any dark matter. In the so-called
s MOdified Newtonian Dynamics (MOND), the rotation and, hence, they were not really new.
a AnintrinsicscatterintheBTFrelationwouldbediffi-
velocityofanisolatedgalaxyis determinedbyits visible
[ cult to accommodate in MOND because, in this theory,
(baryonic) mass. MOND has provensuccessful in repro-
the BTF relation is a direct consequence of the effective
1 ducing the shape and amplitude of a significant fraction
forcelaw. Onthecontrary,theΛCDMparadigmpredicts
v of spiral galaxies (without any dark matter) with only
7 the mass-to-light ratio of the stellar disk as adjustable a real scatter albeit small (Desmond 2012). Hence, the
4 parameter (see Sanders & McGaugh 2002 for a review; BTF relation can in principle be used to distinguish be-
8 Milgrom& Sanders 2007;Sanders & Noordermeer 2007; tweenboth scenarios. For now,it is unclear whether the
5 Swaters et al. 2010). smallscatterinthe BTFrelationisa proofofMONDor
. it is a selection effect. Following this line of arguments,
1 In this paper we will consider gas-rich dwarf galaxies
we have carried out a search for gas rich galaxies that
0 totestMOND. Thistype ofgalaxiesprovidea goodtest
separate from the BTF relation. If MOND is correct,
3 for MOND because (1) in most of these galaxies the in-
all these galaxies should have an obvious problem with
1 ternal acceleration is below the threshold and (2) their
the observed rotation curve like uncertain inclinations
: mass is dominated by gas and hence the predicted rota-
v and distances, or the presence of non-circular motions.
tioncurveisnotsensitivetotheassumedstellarmass-to-
i In particular, we are interested in galaxies that rotate
X lightratioΥ . IntheMONDframework,theasymptotic
⋆ moreslowlythanMONDpredicts,sothattheamplitude
velocity,thatis,therotationvelocityintheoutermostre-
r
of their rotation curves cannot be fitted by arbitrarily
a gionswhererotationcurvestendtobeflat,is(GMa )1/4,
0 increasing the mass-to-light ratio of the stellar compo-
where M is the total baryonic mass of the system. For
nent or by assuming additional undetected matter. The
a sample of 47 gas rich galaxies selected by a strict cri-
selected galaxies might also provide a good target for
teria, McGaugh (2012) finds a tight empirical relation
studying the issue of a link between the levelof stability
between detected baryonic mass and rotation velocity
of galaxies in MOND and the star formation (S´anchez-
in the outermost measured regions, the baryonic Tully-
Salcedo & Hidalgo-G´amez 1999).
Fisher(BTF)relation,asthatpredictedbyMOND.The
The paper is organized as follows. In 2, we give a
low scatter of the data points relative to their errorbars §
brief description of MOND in galaxies. The selection
led McGaugh (2012) to argue that all deviations from
procedure of galaxies and the outcome of the search are
the BTF line can be explained by observational uncer-
presented in 3. Section 4 contains a more exhaustive
taintiesalone. ThisprovidesacheckofMONDwithzero §
studyfocusingonsometestcases. Conclusionsaregiven
in 5.
1Instituto de Astronom´ıa, Universidad Nacional Aut´onoma §
de M´exico, Ciudad Universitaria, 04510 Mexico City, Mexico;
[email protected]
2Departamento de F´ısica, Escuela Superior de F´ısica y
Matema´ticas,IPN,U.P.AdolfoL´opezMateos,C.P.07738,Mex- 2. MOND:BASICEQUATIONS
icoCity,Mexico
2
The LagragianMOND field equations lead to a modi- the rotation curves. However, a test of MOND based
fied version of Poisson’s equation given by onthe capability ofreproducingthe fine structure ofthe
rotationcurvesisamoredelicateissue. Indeed,thereex-
∇Φ
∇ µ | | ∇Φ =4πGρ, (1) istsahandfulofgalaxiesthatMONDdoesnotprovidea
·(cid:20) (cid:18) a0 (cid:19) (cid:21) good fit to the shape of the rotation curve (e.g., Lake &
Skillman 1989; Bottema et al. 2002; Sa´nchez-Salcedo &
where ρ is the density distribution, Φ the gravitational
Lora2005;Corbelli& Salucci2007;Swatersetal.2010).
potential, a is a universal acceleration of the order of
0 However, this fact does not necessarily rule out MOND
10−8 cm s−2. The interpolating function µ(x), with
because of uncertainties in distance and inclination of
x= ∇Φ/a , has the property that µ(x) =x for x 1
| | 0 ≪ the galaxies,beam smearing,non-circularmotions, mor-
and µ(x) = 1 for x 1 (Bekenstein & Milgrom 1984).
≫ phologicalasymmetries,correctionsforasymmetricdrift,
Brada & Milgrom (1995) showed that, to a good ap-
warps and uncertainties in the photometric calibration
proximation, the real acceleration at the midplane of an
(e.g.,Swaterset al.2010). Moreover,the simple relation
isolated, flattened axisymmetrical system, g, is related
between g and g given in Eq. (2) has only been tested
with the Newtonian acceleration, g , by: N
N withtheunderlyingassumptionofaxisymmetryandmay
g inducesomeerrorwhenestimatingtheMOND-predicted
µ | | g =gN. (2) rotation curves in non-axisymmetric galaxies.
(cid:18)a (cid:19)
0 Another source of uncertainty in the exact shape of
Thetwomostpopularchoicesforthe interpolatingfunc- the predicted rotationcurve is the specific formadopted
tion are the “simple” µ-function, suggested by Famaey for µ, which is important at intermediate galactic radii,
& Binney (2005), wherethetransitionbetweentheNewtonianandMOND
regimes takes place. In our selection procedure, we will
x
µ(x)= , (3) evaluate Γobs and ΓM at the outer parts, where the pre-
1+x dicted Γ -value is not sensitive to the exact formof the
M
and the “standard” µ-function interpolating function µ.
x 3.1. Selection procedure and computation of Γ
µ(x)= , (4) obs
√1+x2 In order to check if all galaxies with ‘slow rotation’
presentlargevaluesofΓ,wehaveestimatedΓ formore
proposedbyMilgrom(1983). Unlessotherwisespecified, obs
than80galaxieswithpublishedrotationcurveswhosero-
we will take a = 1.2 10−8 cm s−2 and use the simple
0 × tationvelocities,atthelastmeasuredpoint,arebetween
µ-function (Famaey et al. 2007;Sanders & Noordermeer 35 and 80 km s−1. Galaxies with rising rotation curves
2007; Weijmans et al. 2008). Since we are interested in
in the outermost measured regions were also included.
dwarf galaxies whose dynamics lies in the deep MOND
We have discarded galaxies with circular velocities <35
regime (that is, x = g/a0 ≪ 1), our results are not sen- kms−1 atthe lastmeasured-pointtoavoidthe inclusion
sitive to the exact form of the interpolating function.
of rotation curves with large uncertainties due to asym-
metric drift corrections (e.g., IC 1613 and UGC 7577).
3. GAS-RICHDWARFGALAXIESWITH“LOWROTATION”
On the other hand, we restrict to galaxies with circular
Define Γ(R) as the ratio between the real acceleration velocities < 80 km s−1 because they are more likely to
g and the Newtonian acceleration gN at galactocentric be gas-rich galaxies (e.g., McGaugh 2012) and their dy-
radius R, that is, Γ(R)=g/gN =vc2/(vc2,∗+vc2,g), where namicslie in the deepMOND regime. We selectedthose
vc isthe observedcircularandvc,⋆ andvc,g aretheNew- galaxies with low Γ-values, i.e. galaxies with Γobs <5.5.
toniancontributionsofthestarsandthegastothe rota- In order to estimate v , the gas mass was taken as
c,g
tion curve, respectively. At large enough galactocentric 1.4 times the Hi content to correct for the presence of
radii, Γ is a measure of the ratio between the dynami- He and metals. The typical uncertainty in the Hi mass
cal and the baryonic mass. According to Eq. (2), if we is less than 5% (except for E364-G029which is of 25%).
know vc(R), MOND predicts Γ(R) as ΓM = 1/µ(g/a0), Hence, most of the uncertainties in vc,g comes from the
where g = v2/R. The predicted value Γ can be con- content of molecular gas and other forms of gas. In or-
c M
fronted with the observed value Γobs. If MOND is cor- der to estimate vc,∗, the stellar disk contribution to the
rect, Γ =Γ . rotation curve, we convert luminosity to mass using the
M obs
In the outer parts of dwarf galaxies, MOND predicts models of Bell & de Jong (2001). The (B R) color
−
high Γ-values for galaxies with low rotation velocities. was determined from the magnitudes of the galaxies as
For instance, for a galaxy with v 40 km s−1 at a ra- given in NED. When available, the 25th isophote mag-
c
≃
dius of 7 kpc, MOND predicts Γ 15. In the present nitude was preferred. In Column 7 of Table 1 we give a
M
≈
study,wewillevaluateΓ intheouterpartsofgas-rich range of the stellar mass-to-lightratios in the blue band
obs
galaxies and select those having the smallest values of ΥB. The range is so ample that it includes values previ-
⋆
Γ . Doingso,wewillabletoidentifypotentialgalaxies ously reported by other authors, as well as the bursting
obs
for which MOND could fail to reproduce the observed model, which is the model that best describes most of
amplitude oftheirrotationcurves. Theselectedgalaxies the properties of these galaxies. Note that the stellar
will deserve a detailed analysis of all the available rota- mass-to-lightratios are very dependent on the color and
tion curve data. onthe model. However,since the gasmass dominates in
To be completely successful, MOND should account these galaxies, uncertainties in the stellar mass-to-light
for not only the amplitude of the rotation curves at the ratio do not have a strong impact on the estimates of
outskirt of galactic disks but also the detailed shape of the totalbaryonicmass orin Γ (see columns 8 and10 in
Slowly-rotating galaxies 3
Table 1). curve may be well in accordance to MOND predictions.
Hence, ESO 215?G009 must be excluded.
3.2. Search outcome: galaxies with low Γ Uncertainties in the distance D may also play a role.
obs
ConsidernowhowΓ depends onthe adopteddistance
3.2.1. General comments obs
to the galaxy. Assuming that the Hi disk is infinitelly
Thesearchturnedoutonlysevengas-richgalaxieswith thin, v2 D. The stellar contribution v2 is also pro-
Γobs < 5.5: NGC 3077, NGC 2366, UGC 4173, NGC portionca,gl∝to D if the stellar mass-to-lightc,∗ratio is kept
4861,ESO245-G05,ESO364-G029andHoII.The main fixed. Therefore, Γ (v2 + v2 )−1 D−1. The
properties of these galaxies are compiled in Table 1; the obs ∝ c,g c,∗ ∝
variation of Γ to changes in the adopted distance is
M
rotational velocities at the outer parts, V , are given
rot also simple to derive. Suppose a galaxy that rotates at
in column 9, the empirical values of Γ for the seven se-
velocity V at a radius r . In the deep MOND regime,
rot m
lectedgalaxiesaregivenincolumn10(denotedbyΓobs), we have that µ g/a = V2 /(r a ) at r . Since r
whereas the values predicted by MOND, Γ , are pro- ≃ 0 rot m 0 m m
M scales as r D, we obtain Γ = 1/µ r D (pro-
vided in column 11. The uncertainty in ΥB was also m ∝ M ∝ m ∝
⋆ vided that the distance change is small enough that the
treated as an additional source of error. However, the
outer galaxy is still in the deep MOND regime).
most important error source in the quoted values of Γ
Can uncertainties inherent to such a sample of galax-
is due to uncertainties in the Hi rotation curve of these
ieslikeuncertaintiesintheinclinationsanddistancesex-
galaxies. Figure 1 shows the predicted MOND Γ-value
plain the discrepancy between Γ and Γ ? In the fol-
obs M
and the observed one for these galaxies.
lowing we will try to answer this question. Before mak-
For NGC 3077 and NGC 2366, the values of Γ and
obs ing any further interpretation, it is convenient to briefly
Γ are consistent within the 1σ uncertainty. The Hi
M comment on individual galaxies.
kinematicsofNGC 3077is highlyperturbedbythe tidal
interaction with M81 and M82 and, therefore, is not 3.2.2. Comments on individual galaxies
a good canditate to test MOND. On the other hand,
whereas the inclination of NGC 2366 (i = 63◦) is ade- The MOND rotation curve fitting of UGC 4173 was
alreadystudiedbySwatersetal.(2010). Usingthestan-
quate to accurately estimate the rotation curve, the un-
dard µ-function, MOND overestimates the circular ve-
certainties in the rotation curve are very large. There-
locity in all the points beyond 7 kpc. If the inclination
fore, these two galaxies should be discarded until more
istakenasafreeparameterinthe fits,the MONDcurve
precise data is available. For the remainder five galax-
and the data agrees well for an inclination of 24◦ 5◦.
ies, MOND predicts too large a value of Γ and, hence, it Swaters et al. (2010) argue that an inclination of 2±5◦ to
appears to fail for those galaxies with small Γ .
obs 30◦ is consistent with the Hi morphology because this
Note that the derived values of Γ and Γ depend
obs M galaxy has an optical bar with a faint surrounding disk.
on the distance to the galaxy and on the adopted incli-
NGC4861isairregulardwarfgalaxywithaluminosity
nation. The error bars quoted in Table 1 do not include
uncertainties in the inclination angle or distance to the LB =7.9×108L⊙ andshowsnoevidenceforspiralstruc-
ture(vanEymerenetal.2009b). Thisgalaxywasstudied
galaxy.
in Hi by Wilcots et al. (1996), Thuan et al. (2004) and
Consider first uncertainties in the inclination angle.
Beyond a few disk scale radii, where the mass distri- van Eymeren et al. (2009a,b). For ΥB⋆ = 0.3M⊙/L⊙,
butionhasessentiallyconverged,v andv donotde- the total baryonic mass is 9.1 108M⊙. It is one of
c,⋆ c,g ×
pend on the adopted inclination. The amplitude of the the most inclined galaxies in our selected sample. The
rotation curve at the outer parts V , depends on the optical inclination from HYPERLEDA is 90◦ (Paturel
rot
inclination i of galaxy and changes as V (1/sini). et al 2003). Thuan et al. (2004) derived an inclina-
For modest changes in the inclination sorottha∝t the outer tionof82◦ forthe outermostHitiltedring,whereasvan
dynamics lie in the deep MOND regime, it holds that Eymeren et al. (2009a,b) estimated a mean Hi inclina-
Γ V−2 sin2i, whereas Γ V2 (1/sin2i). tionof65◦ 5◦within7kpc. Usingthelatterinclination,
M ∝ rot ∝ obs ∝ rot ∝ we infer Γ± =4.7 at 7 kpc.
Inordertoavoidstrongeffectsonthedeterminationof obs
Tilted-ring fits for ESO 245-G05 give an inclination
the rotation curve, Begeman et al. (1991) required that
thegalaxiesshouldhaveHiinclinationsbetween50◦and of 54◦ 10◦ (Coˆt´e et al. 2000). Its Hi mass is 2
± ×
80◦. On the other hand, McGaugh (2012)demands con- 108M⊙. Adopting ΥB⋆ = 1, the total baryonic mass is
sistency between optical and Hi inclinations. Whereas 4.4 108M⊙. Atthe lastmeasuredradiusof3.5kpc, we
×
ESO 245-G05,NGC 4861and ESO 364-G029have com- find Γobs =3.0 0.6, which is a factor of 2 smaller than
fortableHiinclinationanglesbetween50◦ and80◦,none the predicted va±lue by MOND. Whereas this is a poten-
of the selected galaxies comply with McGaugh’s consis- tial galaxy to test MOND, it is still premature to make
tency criterion. anyconclusiongiventhelargeerrorsquotedontheincli-
ThegalaxyESO215?G009exemplifiesthe importance nation and its uncertain kinematics due to the presence
ofhavinggalaxieswithhighinclinationsinordertoame- of a strong bar.
liorate the uncertainties in this parameter. Warren et HoII is an irregular galaxy in the M81 group. Careful
al. (2004) report a circular velocity of 51 8 km s−1 at analyses of Hi observations of HoII have been carried
7 kpc for an inclination i = 36◦ 10◦. ±However, van out by three independent groups: Puche et al. (1992),
Eymeren et al. (2009a), using a ±smaller inclination of Bureau & Carignan (2002) and Oh et al. (2011). The
28◦, derive a remarkably larger amplitude of the rota- Hi rotation curves derived by all these authors are very
tion curve of 75 km s−1 at the same angular distance. similar. Ohetal.(2011)derivedthestellarmass-to-light
Given the low inclination and the uncertainties in the ratio in K-band, ΥK, for HoII. With such a ΥK-value
⋆ ⋆
distance to this galaxy, the amplitude of the rotation (which we will refer to as the nominal value), the stellar
4
mass in the disk of HoII is of 1.5 108M⊙. Given that ated uncertainty 5◦, indicating that this inclination is
the total mass in gas is 9 108M⊙×(Bureau & Carignan unlikely. ∼
×
2002), the total (gas plus stars) baryonic mass in HoII
is about 10.5 108M⊙. With these empirical values, we 4.2. The galaxies HoII and ESO 364-G029
infer Γ =2× 0.4 at a radialdistance of7 kpc. This is 4.2.1. A comparative study
obs
±
muchsmallerthanthe valuepredictedbyMOND,which
The galaxies HoII and ESO 346-G029 have similar
is about 19.
baryonicmassbut the amplitudes beyondgalactocentric
ESO 364-G029 has a luminosity of LB = 6.6×108L⊙ distances of R=5 kpc are rather similar. In the case of
for a distance of D = 10.8 Mpc (Kouwenhoven et
HoII, the method of Oh et al. (2011) minimizes the ef-
al. 2007). The models of Bell & de Jong (2001) pre-
fect of non-circular motions. The rotation curve of HoII
dict a stellar mass-to-light ratio of 0.2–0.6 in the blue
Hbaindd,isitmripbluytiinognaisstemllialdrlmyaasssyomfm(1e.t∼5ri−c4a)n×d1r0o8uMgh⊙l.yTfohle- wtciooarnsrecwcotarisorenncsottewdmoufaolddreacfsaoyurmsEemSaeOtbr3oic6o4sdt-rGiof0tf2aw9.hfeeAwrseyakmsmmthse−ist1rcicionrdrtrehicfet-
lows the stellar brightness distribution. The Hi mass is
circular velocities of these galaxies.
6re.4ac×h1es08aMv⊙aluaendofth4e2rkomtatsi−o1navtelaocdiitsyt,aanscseumofin7gkip=c.7D0e◦-, Remind that the asymptotic velocity is defined as
(GM a )1/4, with M the baryonic mass. Using the
spitetheasymmetricappeareanceoftheHi,therotation bar 0 bar
estimates of the baryonic masses given in Table 1 and
curve is fairly symmetric. The corresponding Γ -value
obs a =1.2 10−8 cm s−2, MOND predicts an asymptotic
at R=7 kpc is 2±10..05. ve0locity o×f 63 km s−1 for HoII and 66 km s−1 for ESO
We conclude that, at present, the galaxies NGC 4861,
364-G029. Thepredictedrotationalspeedisinexcessby
HoII and ESO 364-G029 are promising targets to test
25 km s−1 in HoII and ESO 364-G029. In the following,
MOND.Fortheremainderfourgalaxies,furtherdecreas-
we will concentrate on the case of HoII because the ro-
ing systematic uncertainties in these galaxies could pro-
tationcurvehasamuchbetterspatialresolutionthanin
vide a strong test to the MOND framework. In the
ESO 364-G029. In addition, the irregular and lopsided
nextsection,wewillconsiderthewholeavailablerotation
morphology of ESO 364-G029 could induce systematic
curveofthegalaxiesNGC4861,HoIIandESO364-G029
biases if one assumes axisymmetry.
to quantify the difference between predicted circular ve-
locities and measured velocities. The galaxies HoII and 4.2.2. MOND in HoII
ESO 364-G029 are expected to be the most problematic
Since HoII is embedded in the external gravitational
because they present the lowest Γ-values (Γ 2).
obs
≃ field of M81 group, one has to quantify the external
4. ROTATIONCURVES:NGC4861,HOIIANDESO364-G029 field effect (EFE) described in Bekenstein & Milgrom
(1994), by comparing the internal and external accel-
4.1. The galaxy NGC 4861
erations. Assuming that the M81 group is bound, the
The left panel of Figure 2 shows the observedrotation external acceleration is 0.7 10−10 cm s−2 (Karachent-
curve of NGC 4861 from van Eymeren et al. (2009b) sev et al. 2002), which is m×uch smaller than its internal
togetherwiththe predictedMONDrotationcurve(solid accelerations( 6 10−10 cm s−2 and2 10−10 cm s−2
lines),forthesimpleandthestandardinterpolatingfunc- at R=7 kpc a∼nd R×=20 kpc, respective×ly). Thus, EFE
tions. The adopted distance was D = 7.5 Mpc. We should be small at R<10 kpc.
have modelled only the inner 7.5 kpc from the dynamic Figure 3 shows the predicted HoII rotation curve in
center because beyond this radius the observed rotation MOND under various ΥK assumptions [the nominal
⋆
curve is affected by uncertainties caused by the sparsely valueasderivedbyOhetal.(2011),“minimumdiskplus
filled tilted rings. For the stellar disk, we have assumed gas”,and twice the nominal value]. The “minimum disk
ΥR⋆ =0.3M⊙/L⊙. WeseethatMONDoverestimatesthe plus gas” includes the gas component and uses the min-
observed rotation velocities at any radius. imum value of Υ compatible with the requirement that
⋆
If the distance and ΥR are left as free parameters, an thetheoreticalcircularvelocitymustbepositiveandrea-
⋆
acceptable fit is obtained for D = 4.2 Mpc and ΥR = sonably smooth. The discrepancy between the observed
⋆
0.12 (see Fig. 2). The distance to this galaxy has been andthepredictedrotationcurvesisapparent. Theeffect
determined with different methods and all gives D > 7 ofvaryingΥ ortheinterpolatingfunctionontheMOND
⋆
Mpc. Using the Virgocentric infall model of Schechter circular velocity at the outer disk is small, less than 10
(1980) with parameters γ = 2, v = 976 km s−1, km s−1 at R = 7 kpc. For illustration, Figure 4 shows
Virgo
w⊙ = 220 km s−1 and a Virgo distance of 15.9 Mpc, a the combined rotation curve from Bureau & Carignan
distance of 10.7 Mpc is derived for NGC 4861 with an (2002) and Oh et al. (2011) for the updated HoII dis-
uncertainty of 20% (Heckman et al. 1998; Thuan et tance of 3.4 Mpc. Note that at R > 10 kpc, velocities
al. 2004). Thus∼, it is unlikely that changes in distance were only measured in the approaching side.
alonecan explainthe discrepancybetweenthe predicted The value of a0 is universal but needs to be fixed
and the observed rotation curves. from observations. Even adopting a value of a0 at the
The MOND fit to the rotation curve can be improved lower end of the best-fit interval derived by Begeman et
itfhethfiet.incIlfi,niantsiotenadofotfhaedgoapltainxyg athnednΥoR⋆mianrael lienfctlifnraeteioinn sa−l.2(,1t9h9e1M) aOnNdDGecnirtciulelaertraolt.a(t2io0n11s)p,eae0d=is0o.n9ly×a10fe−w8kcmm
i = 65◦ derived by van Eymeren et al. (2009b), we use s−1 slower.
an inclination of i = 43◦, the MOND curve and the de- In the MOND prescription, the amplitude of the rota-
rivedrotationcurveareconsistent. However,the change tion curve can be accounted for by adopting a distance
required in inclination is much larger than the associ- to HoII of 1.5 Mpc and a = 0.9 10−8 cm s−2 (see
0
×
Slowly-rotating galaxies 5
Figure 5). Given that the uncertainty in the distance is not be fitted by arbitrarily increasing the mass-to-light
of 0.4 Mpc (Karachentsev et al. 2002), this likely indi- ratioof the stellarcomponent orby assumingadditional
cates thatMOND cannotbe made compatible with that undetected matter.
rotation curve by a reasonable adjustment of galaxy’s Anamplitudeoftherotationcurvelowerthanexpected
distance. couldbe causedby an overestimateofeither the inclina-
A more delicate issue is the error resulting from the tion or the distance to the galaxy. In order to quantify
uncertainty in the inclination of the galaxy. The optical theseeffects,wehavefocusedontwogalaxies: NGC4861
inclination from LEDA is of 45◦ (Paturel et al. 2003). andHoII.Forthesegalaxies,wefindthatthediscrepancy
Here, we haveused a globalinclination of the Hi disk of between the observed and the predicted rotation curves
40◦ intheinnerparts(Ohetal.2011),and84◦ fortilted inMONDcannotbeaconsequenceofadoptinganincor-
ringsatR>12kpc (Bureau&Carignan2002). Itturns rectdistancebecauseitisunlikelythatthe distancesare
outthat if the inclinationis takenas a free parameter,a uncertain by that much.
meaninclinationof25◦ wouldyielda circularvelocityof An inclination of 25◦, instead of 40◦, is required to
60 km s−1 at R = 7 kpc. In a recent posting during maketherotationcurveofHoIIcompatiblewithMOND.
∼
thecourseofsubmittingthispaperandmotivatedbyour It is clear that the main source of systematic uncertain-
preprintarXiv:1105.2612,Gentileetal.(2012)re-analyze tiesinthedeterminationoftheamplitudeoftherotation
therotationcurveinHoIIbymodellingitsHidatacube curve in HoII is its low inclination. The strategy is to
and find that the inclination is much closer to face-on look for a galaxy as similar as possible to HoII but with
than previously derived. At this lower inclination, the a higher inclination to reduce geometrical uncertainties.
rotation velocity becomes commensurate with what is We found that NGC 4861 and ESO 364-G029 rotate at
expected from MOND. a similar velocity than HoII, have similar baryonic mass
In the very outer disk, at R > 12 kpc, an inclination buttheir inclinationsaresignificantlylarger. Inthe case
of 45◦ is requiredto reconcile MOND with observations. ofNGC4861,thechangerequiredininclinationtofitthe
This value is far lower than the one derived by Bureau observed rotation curve in MOND is much larger than
& Carignan (2002) fitting tilted ring models (84◦). We the associated uncertainty. The galaxy ESO 364-G029
conclude that more accurate determinations of HoII in- is very interesting because MOND severely overpredicts
clinationintheouterpartswillprovideamoredefinitive by 50% the observed circular velocity at the outer parts
test to MOND. ofthe disk. Contraryto HoII, it has a comfortable incli-
nation of 70◦. We conclude that deeper photometric
5. FINALREMARKSANDCONCLUSIONS andHiob∼servationsofESO364-G029,togetherwithfur-
ther decreasing systematic uncertainties, may provide a
MONDpredictsatightcorrelationbetweentheasymp-
strong test to MOND.
totic circularvelocity andthe totalbaryonicmass of the
galaxy. Gas-rich dwarf galaxies are an interesting and
unique test of modified theories of gravity. Their inter- We would like to thank Elias Brinks for his encour-
nal accelerations are below a and the stellar mass in agementto make these results public, MargaritaRosado
0
these galaxies is not important in the budget of total for helpful discussions and the anonymous referee for a
mass. useful report. The authors made use of THINGS ‘The
Foralargesampleofgas-richdwarfgalaxieswithrota- Hi Nearby Galaxy Survey’ (Walter et al. 2008). This
tion velocities between 35 and 80 km s−1, we have com- research hasmade use of the NASA/IPAC Extragalactic
puted the parameter Γ , defined as the ratio of real Database (NED) which is operated by the Jet Propul-
obs
to Newtonian accelerations, and the predicted value in sion Laboratory, California Institute of Technology, un-
MOND Γ . We found that five galaxies (UGC 4173, der contract with the National Aeronautics and Space
M
HoII,ESO245-G05,NGC4861andESO364-G029)have Administration. This workwassupportedby the follow-
Γ <Γ . Forthesegalaxies,MONDoverestimatesthe ing projects: CONACyT 165584,PAPIIT IN106212and
obs M
presentlyavailablerotationspeeds, andhence, they can- SIP-20121135.
REFERENCES
Begeman,K.G.,Broeils,A.H.,&Sanders,R.H.1991,MNRAS, Heckman,T.M.,Robert,C.,Leitherer,C.,Garnett,D.R.,&van
249,523 derRydt,F.1998,ApJ,503,646
Bekenstein, J.,&Milgrom,M.1984,ApJ,286,7 Karachentsev, I.D.etal.2002,A&A,383,125
Bell,E.F.,&deJong,R.S.2001, ApJ,550,212 Kouwenhoven, M.B.N.,Bureau,M.,Kim,S.,&deZeeuw,P.T.
Bottema, R.,Pestan˜a, J.L.G.,Rothberg, B.,&Sanders,R.H. 2007,A&A,470,123
2002, A&A,393,453 Lake,G.,&Skillman,E.D.1989,AJ,98,1274
Brada,R.,&Milgrom,M.1995,MNRAS,276,453 Martin,C.L.1998,ApJ,506,222
Bureau,M.,&Carignan,C.2002,AJ,123,1316 McGaugh,S.S.2011,arXiv:1109.1599
Corbelli,E.,&Salucci,P.2007,MNRAS,374,1051 McGaugh,S.S.2012,AJ,143,40
Coˆt´e,S.,Carignan,C.,&Freeman,K.C.2000, AJ,120,3027 Milgrom,M.1983,ApJ,270,365
Desmond,H.2012,arXiv:1204.1497 Milgrom,M.,&Sanders,R.H.2007,ApJ,658,L17
Famaey,B.,&Binney,J.2005, MNRAS,363,603 Oh,S.-H.,deBlok,W.J.G.,Brinks,E.,Walter,F.,&Kennicutt,
Famaey,B.,Gentile,G.,Bruneton,J.-P.,&Zhao,H.2007, R.C.2011, AJ,141,193
Phys.ReviewD,75,063002 Paturel,G.etal.2003,A&A,412,45
Foreman,S.,&Scott, D.2012,Phys.Rev.Lett.,108,141302 Puche,D.,Westpfahl, D.,Brinks,E.,&Roy,J.-R.1992,AJ,103,
Gentile,G.,Famaey,B.,&deBlok,W.J.G.2011, A&A,527,76 1841
Gentile,G.,Angus,G.W.,Famaey, B.,Oh,S.-H.,&deBlok,W. S´anchez-Salcedo, F.J.,&Hidalgo-Ga´mez,A.M.1999,A&A,345,
J.G.2012, A&A,543,47 36
6
Fig. 1.— Γobs (squares) and ΓM (triangles) for the selected galaxies. The galaxies were ordered according to their mean inclination
(abscissa).
Fig.2.—RotationcurveofNGC4861takenfromvanEymerenetal.(2009b),togetherwiththecontributionsofthestellardisk(dotted
lines)andgas(dashedlines)forD=7.5MpcandΥR⋆ =0.3(leftpanel)andforD=4.2MpcandΥR⋆ =0.12(rightpanel). Thefulllines
representtherotationcurveaccordingtoMONDprescriptionusingthesimpleµ-function(uppersolidcurves)orthestandardµ-function
(lowersolidcurves). Herewetakea0=1.2×10−8 cms−2.
S´anchez-Salcedo, F.J.,&Lora,V.2005,inProgressinDark vanEymeren,J.,Trachternach, C.,Koribalski,B.S.,&Dettmar,
Matter Research,ed.J.ValBlain(NovaPublications:New R.-J.2009a,A&A,505,1
York) vanEymeren,J.,Marcelin,M.,Koribalski,B.S.,Dettmar,R.-J.,
Sanders,R.H.,&McGaugh,S.S.2002,ARA&A,40,263 Bomans,D.J.,Gach,J.-L.,&Balard,P.2009b,A&A,505,105
Sanders,R.H.,&Noordermeer,E.2007,MNRAS,379,702 Walter,F.,Brinks,E.,deBlok,W.J.G.,Bigiel,F.,Kennicutt,
Schechter, P.L.1980,AJ,85,801 R.C.,Thornley,M.,&Leroy,A.2008,AJ,136,2563
Swaters,R.A.,Sanders,R.H.,&McGaugh,S.S.2010,ApJ, Warren,B.E.,Jerjen,H.,&Koribalski,B.S.2004,AJ,128,1152
718,380 Weijmans,A.-M.,Krajnovic,D.,vandeVen,G.,Oosterloo,T.
Thuan,T.X.,Hibbard,J.E.,&L´evrier,F.2004,AJ,128,617 A.,Morganti,R.,&deZeeuw,P.T.2008,MNRAS,379,1343
Wilcots,E.M.,Lehman,C.,&Miller,B.1996, AJ,111,1575
TABLE 1
Comparison of the relevantparametersof the selected galaxies
Name D MB LB hii Mgas ΥB⋆ Mbar Vrot Γobs ΓM Source
Mpc 108L⊙ deg 108M⊙ 108M⊙ kms−1
E215?G009 4.2 -12.9 0.23 36◦ 7.1 0.5-2 7.2–7.6 51±5at7kpc 5.6±11..42 10.7±2 1
5.25 -13.4 0.36 28◦ 11.1 0.5–2 11–12 75±1185 at8kpc 9.0±53..05 6.1±31..08 2
NGC3077 3.8 -17.75 19.6 46◦ 12.3 0.2–0.6 16–24 62±2105 at6kpc 3.0±31..75 6.5±42..04 3,4
NGC2366 3.4 -17.17 11.5 63◦ 9.1 0.15–0.3 10.8–12.5 60±16at7kpc 5.4±32..77 8.0±62..06 2,3,5
UGC4173 18.0 -16.50 6.2 40◦ 34.0 0.5–1 37–40 48±5at9kpc 2.8±0.7 15±32..55 6
NGC4861 7.5 -16.76 7.9 65◦ 6.7 0.15–0.3 8.0–9.1 46±52 at7kpc 4.7±10..37 12.9±12..00 2,7
E245-G05 2.5 -15.0 1.6 54◦ 2.8 0.7–1.5 3.9–5.2 43±1at3.5kpc 3.0±0.6 7.8±0.3 8
E364-G029 10.8 -16.6 6.6 70◦ 9.0 0.2–0.6 10–16 42±3at7kpc 2.0±10..05 15.0±21..55 9
HoII 3.4 -16.87 8.7 49◦ 9.0 0.17 10.5 37±4at7kpc 2.0±0.4 19.5±43..55 3,5
Column(1): Nameofthegalaxy. Column(2): Adopteddistance. Column(3): AbsoluteBmagnitude. Column(4): Totalblue-bandluminosity. Column(5): Averagevalueofthe
inclination. Column(6): Totalmassingas. Column(7): Stellarmass-to-lightratiointheblueband. Column(8): Totalbaryonicmass. Column(9): Rotationvelocity. Column(10)
and(11): Γobs andΓM =1/µ(g/a0). Column(12): References,1: Warrenetal.2004;2: vanEymerenetal.2009a; 3: Walteretal.2008;4: Martin1998; 5: Ohetal.2011;6: S
Swatersetal.2010; 7: vanEymerenetal.2009b;8: Coˆt´eetal.2000;9: Kouwenhoven etal.2007. lo
w
ly
-
r
o
t
a
t
in
g
g
a
la
x
ie
s
7
8
Fig.3.—RotationcurveofHoIIfromOhetal.(2011)together withthecontributionsofthestellardisk(dotted lines)andgas(dashed
lines)forthenominalΥ⋆-value(leftpanel),minimumdiskplusgas(centralpanel)andtwicethenominalΥ⋆-value. Thefulllinesrepresent
therotationcurveaccordingtoMONDprescriptionusingthesimpleµ-function(uppersolidcurves)orthestandardµ-function(lowersolid
curves). MONDisunabletoprovidetheamplitudeoftherotationcurve. HerewetakeD=3.4Mpcanda0=1.2×10−8 cms−2.
Fig.4.—CombinedrotationcurveofHoIIasmeasuredbyOhetal.(2011) (atR<7kpc)andbyBureau&Carignan(2002) atR>7
kpc. The dashed line represents the contribution to the rotation curve of the gas disk. The solid line represents the MONDian rotation
curve in the “minimum disk+gas” assumption, using the standard µ-function and a0 = 1.2×10−8 cm s−2. All the variables have been
rescaledfortheadopted distanceofD=3.4Mpc.
Slowly-rotating galaxies 9
Fig. 5.—HoIIrotationcurveinMONDadoptingthestandardµ-functionwitha0=0.9×10−8 cms−2 andadistancetothegalaxyof
1.5Mpc,afactor2.3closerthanthenominaldistance. ThekeytolinesisthesameasinFig.3.